Abstract
We consider a time-independent BÉnard equation (*) -Aw+(λ0 + ε)PMw+N(u,w)+(aθ x+bθy)w+τf = 0 on the infinite layer R2 × [-1/2, 1/2]. Here f = f(z) is an exterior force depending only on z ∈ [-1/2, 1/2], w satisfies Dirichlet conditions in the z-direction and L1, L2-periodic conditions in the x, y-direction, while a, b satisfy a diophantine condition. λ0 is the critical Rayleigh parameter. It is shown that for generic f and small τ, ε, a, b, (*) has solutions w such that (aθx + bθy)w ≠ 0. These solutions give rise to periodic traveling wave solution of the time-dependent version of (*) (with a = b = 0). The proof is via bifurcation methods related to Hopf bifurcation.
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References
S. Chandrasekhar, Hydrodynamic and hydromagnetic stability. Dover Publ., New York 1961.
G. Crandall, P. Rabinowitz: The Hopf bifurcation theorem in infinite dimensions. Arch. Rat. Mech. Analysis 67, no. 1, 53–72, 1977.
A. Friedman, tFoundations of modern analysis. Dover Publ., New York, 1982
T. Kato, Perturbation theory for linear operators. Springer, Berlin, New York, 1980.
K. Kirchgässner, H. Kielhöfer, Stability and bifurcation in fluid dynamics. Rocky Mountains J. of Math., vol. 3, no. 2, 275–318, 1973.
K. Kirchgässner, Bifurcation in nonlinear hydrodynamic stability. SIAM Review, vol. 17, no. 4, 452–483 (1975).
A. Pazy, Semigroups of linear operators and applications to PDE’s. Appl. Math. Sc. 44, Springer, New York, 1983.
P. Roberts, An introduction to magneto-hydrodynamics, Longmans, 1974.
D. Sattinger, Topics in stability and bifurcation theory. Lecture Notes in Math. 309, Springer, Berlin, New York, 1973.
B. Scarpellini, Equilibrium solutions of the Bénard equations with an exterior force. J. Diff. & Integral Eq. 16, no. 2, 129–1958, 2003.
B. Scarpellini, W. von Wahl, Stability properties of the Boussinesq equations. ZAMP 49, 294–321, 1998.
B. Scarpellini, Stability, instability and direct integrals. Chapman & Hall/CRC, Boca Raton, 1999.
W. von Wahl, The Boussinesq equations in terms of poloidal and toroidal fields and the mean flow. Bayreuther Math. Schriften 40, 203–290, 1992.
B. Sandstede: Stability of traveling waves. Handbook of dynamical systems, Vol. 2, (2002) 983–1055, Elsevier.
D. Chae, P. Dubovski: Traveling wave-like solutions of the Navier-Stokes and related equations. J. Math. Anal. Appl. 204 (1996), 930–939.
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© 2005 Birkhäuser Verlag Basel/Switzerland
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Scarpellini, B. (2005). Bifurcation of Traveling Waves Related to the Bénard Equations with an Exterior Force. In: Brezis, H., Chipot, M., Escher, J. (eds) Nonlinear Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 64. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7385-7_26
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DOI: https://doi.org/10.1007/3-7643-7385-7_26
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